Except that heavy armour penalizes your stealth rolls and lowers your movement speed if you don't have enough strength. If people are using encumbrance then that suit of armour would also limit their carry capacity. Still, I'm not sure it would matter too much, if it did then I'd expect to see more people picking up dragon sorcerer, barbarian, or monk for their armour bonuses.
It's negligible in many cases. Ring mail (AC 14) has no Strength requirement. While it carries the Stealth penalty, a character who dumped Dex (which is who it ought to appeal to most) isn't likely to be relying much on stealth to begin with. As for carrying capacity, it only matters if you track encumbrance (some groups don't) and then only until you get a bag of holding or similar item. Most casters (the characters this would most appeal to) don't need much in the way of equipment to begin with.
Then there's everything you give up to gain such an ability. As I said, this would primarily appeal to arcane casters (since most other classes can get as good or better AC). Being a level behind in obtaining higher level spells is a painful choice (at least, it was for me when I played my cleric/wizard, although I went knowledge domain and took the heavy armor feat). I made the choice primarily for RP reasons, but I was constantly thinking about the spells I could have been casting if I hadn't dipped cleric. That's somewhere around 2 sessions where you could be casting polymorph (for example) if you hadn't opted for a better AC. For something that, if you play smart and stay behind the front line, won't come up all that often (although when it does, you may be glad you did).
Then there are the RP considerations. For a devout character, it might make sense. For more... scheisty... character concepts it typically won't (the trickster domain might make sense for such characters, but it doesn't grant heavy armor).
Which is why I consider it a minor boost that certainly not all players would opt into. It's already an option, and most players ignore it. While this change removes some of the drawbacks, those are fairly minor in many cases to begin with.
Now, I'm not saying that every DM should allow this. Simply that if you don't like the idea of heavily armored clerics, then this is a simple way to implement that change.
Edit:
You should take your players into consideration before making such a change. If you have players who make leveling decisions based on system mastery, then it's possible that this solution isn't well suited to your group. If you have old school players who envision their character as the iconic, heavily armored cleric then this probably isn't a good solution for you. On the other hand, if your players are primarily motivated by RP choices, I can't imagine you'd have anything to worry about even if you decide on the most powerful option (just granting AC 18 from the get go).